Sunday, November 22, 2015

This is to figure out how to assemble the Fibonacci Numbers in 4 stages, ie over 4 rows.

please click on images to see them more clearly, and in sequence. thank you.

Thus, to make 1597 from 987,
increase every 6th loop as required in description, to get 1140.
increase every 7th loop as described to get 1292
then increase every 8th loop to get 1445
then increase every 9th loop to get 1597.

note : there are 76 or 77 stitches at ends where there is no increase.

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All the above looks as if the Fibonacci Series has a magical elegant secret within it.
The numbers are so eerily exact! To many decimal points!
And then It occurred to me that there is a natural Reason.
It is a function of the mathematical action. As follows.

This is what it looks like when one crochets it; one row for each Number.
To eliminate the outer frill one would have to make non decreasing rows in between the KNITTING.
Begin with cast on 610 stitches, with fine yarn is best. My red here is thicker 5 ply....bulky. The blue and white and green are 4 ply cotton from my yarn stash.

Please click on images to see more clearly. Thank you.

This is more like the real true proportions of Earth.
The living section from deepest ocean trench up to the top of atmosphere would be a mere few hairs' breadths.

One reflects that Earth is so precious. It is unthinkable that we humans might destroy it all.

Dr Michel Odent has written some books on this, and I discuss the topic in my blog http://www.acolourfulpost.blogspot.com

Maybe hardly anyone reads this blog. Is Google putting a block on some Internet material? One day I searched for "a colourful post" blogspot, and nothing of mine came up!

Interesting. This shows that the best way is to round off the numbers to a whole number ie the lower value (not .5 ). Increases (or decreases) may be made 1) every 3 stitches ie 3 to 4 stitches; 2) every 4 stitches ie 4 to 5 stitches. Adjust the remainder of stitches needed at the end of the row.

Please click on Tables to enlarge image. You can then right click and save, copy onto Word, or save link as...a download or favourite.

This is new. An Equation:-
please click on image to enlarge it, to see it better.

I have been most excited at these discoveries!Previously it was trial by error to get the numbers right, but this is exact! except for the very small numbers at the beginning of the series where we must figure it out simply.

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Imaginary Fibonacci Ripples, The blue is a Fibonacci number, the white is the half Fibonacci number !So, the Pattern is to increase every 3rd stitch to make 4, for the half Fibonacci numbers. Increase every 4th stitch to make 5 for the real Fibonacci numbers. Here we have calculated the radius of each circle with circumference equal to the Numbers and the distance between each band. At first I crocheted the last stitch by "hook into loop, crochet 2 chains, Begin new row. Single crochet was used ie hook into loop, wool around hook, pull wool or cotton through, wool around hook, pull through all stitches. There was an ugly join. A piece of colored yarn was used to keep track of the end of the row. Often I had to count to check if it was ok.To make a clean join i just continued on with the spiral, and only the colored yarn marked the end of the row.

*** Please note that UK Europe Australia crochet instructions and gauges are different from American usage. Please see post of 12/26/2014, "Information for US and UK Crocheters"******************************************

Calculations
for Concentric Circles (Ripples)

iaw the
Fibonacci Series and half Fibonacci Series.

Fn radius r1
– r2 rows

2584 411.3 78.5 79

2091 332.8 78.6 79

1597 254.2 48.6 49

1292 205.6 48.5 49

987 157.1 29.9 30

799 127.2 30.1 30

610 97.1 18.5 19

494 78.6 18.6 19

377 60 11.5 12

305 48.5 11.4 11

233 37.1 7.2 7

188 29.9 7 7

144 22.9 4.3 4

117 18.6 4.4 4

89 14.2 2.7 3

72 11.5 2.7 3

55 8.8 1.7 2

45 7.2 1.7 2

34 5.4 0.9 1

28 4.5 1.2 1

21 3.3 0.6 1

17 2.7 0.6 1

13 2.1 0.5 1

10 1.6 0.3 0

8 1.3

I have tried another way to upload the Table- control c to copy document, the control v to paste here.

You can see on the blue and white model, that the last white band has 11 white rows, this shows it has circumference ...305.

I am unable to see it right! Very stressful; I am working at a friend's place to upload all of this; no electricity or Internet at home in my caravan. I shall go away and check on it.

A story of me in High School. Maths class. Geometry. early 1960s. All those of us who got a Theorem wrong in a Test were humiliated; forced to line up in the middle of the gymnasium at lunch time, for the full hour. I was having a rough time, unable to see the way the geometry worked. For the life of me, I could not get it. Usually I came top in Biology and such, but this day I had problems. No wonder I am fated to do blogs on Geometry!

You can click on the images to see them more clearly.Right click to save or print.....From this info you can crochet a flat model of ripples in this fashion, increasing every 3rd stitch and then no increase for the required number to attain the half Fibonacci number iaw the Tables. Increase every 4th stitch to make 5 etc to make the Fibonacci number.eg last band, blue is 377 from 305 white. column 3.base is 305 ie 72 x 4 = 288 + 17;increase is 72 x 5 = 360 +17.Therefore increase every 4th stitch 72 times until 17 loops remain, and crochet normally the last 17.So the same for all the Series, as above. Good luck. Any mistakes you might find on my blog, please let me know. I am always editing and correcting!*********************a photo still to come...

insert 19/11/2015

This Table is a correct one. Some previous info has been rounded off to whole numbers, to make it easier to knit or crochet.

The description "eternal protocols" describes the essential Patterns we can use when crocheting or knitting the Fibonacci Series. It is remarkably distinct and I was excited to discover this for myself.

This is the correct Table.

For example. Take 610 to 987.
We need to add 377 to 610. Divide 610 by 377 = 1.61803....
So we have to increase a certain number of stitches.
This was figured out in a different way back in 2010.
the Equation we found was F(n) = 2 x F(n-2) + F(n-3). and it holds here.

This is a simplified Pattern for knitting or crocheting.

Cast on 610 stitches, knit in accordance with Pattern, you can put one plain row in between each decrease row.Just make sure the balance is kept, with 2x stitches in the middle. When knitting it depends how many stitches fit on the knitting needle; work is shaped by decreasing- working the equation backwards. This green work is done in fine crochet cotton.You can crochet beginning with small numbers.------------------------------------------------------------------------There is a second way to build the Fibonacci Series, from 2010; the Vortex.....unit is 4 to 7 or 13 to 21 as follows:-

Please click on image if you need to enlarge it.

Also, if you wish to copy it, click on it and then right click. Save as.....doc, or copy onto Word.
Or save link as....put it in downloads or favourites.

NB. There is a third way to assemble the fibonacci Series in knit or crochet and it involves the Equation using number 9 or 18; work is still to be developed.
Reference is 26/5/2011 The Giant Clam.
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This one is 10946 ! Crocheted using Equation Fn = 2F(n-2) + F(n-3) as in first description. It was published 5/3/2011 in Horns, tusks, fangs and thorns, spines ans prickles .

This one is also 10946; placed on a smooth surface, it naturally conforms into a 2 hemisphere, a brain-like shape.

---------------------------------------------Instructions for Making a Model of a Spiral "snail like" Shape.

About Me

Born in Germany January 1946 and migrated to Australia in 1949.
B.Sc from UNSW 1970.
Name is Tiiu Vanamois.
Email is tmvanam@gmail.com .
Blogs are www.awhatifworld.blogspot.com
and www.acolourfulpost.blogspot.com
and www.biomathcraft.blogspot.com.
-------The first is to do with my Land and ecology of the World.----The second is about Art and Health.----The third is mainly a study of Mathematical Series, namely Fibonacci, Lucas, Bifurcating Series. Plus varius related themes--------- Of big interest is the making of models using crochet or knitting ------if anyone is interested in working with this, please do so ------------There are two asterisks **, after all posts in date index to do with crochet or knitting or making patterns for models-------------------- 4 asterisks** ** for Tables.